Donaldson-Thomas theory was first designed to count sheaves on Calabi-Yau threefolds. In [1]}, Thomas defined the numerical Donaldson-Thomas invariants, giving the virtual Euler number of the moduli space of stable coherent sheaves on a Calabi-Yau threefold, using the perfect obstruction theory given by Serre Duality on the Ext spaces of the sheaves. In [2]}, Behrend gave a new interpretation of these invariants: expressing the moduli space locally as the critical locus of a potential, the numerical Donaldson-Thomas invariant is given by an Euler number weighted at each point by the Milnor number. In [3]} and [4]}, Kontsevich and Soibelman have sketched the definition of a cohomological refinement of this counting, with value in the abelian category of monodromic mixed Hodge modules (MMHM), using the functor of vanishing cycles of this potential, in a partially conjectural framework.
